Question

Simplify this question

Answer 1

Qᴜᴇsᴛɪᴏɴ :-

Simplify :- 1/√[ 7 - 4√3]

Sᴏʟᴜᴛɪᴏɴ :-

→ 1/√[ 7 - 4√3]

Lets Try to Solve Denominator Root Part First ,

→ 7 - 4√3

Splitting The Terms, we can write ,

→ 4 + 3 - 2 * 2 * √3

→ (2)² + (√3)² - 2 * 2 * √3

comparing it with a² + b² - 2ab = (a - b)² we get,

→ (2 - √3)²

So,

→ √[2 - √3]²

we know that, (√a)² = a ,

→ (2 - √3)

Now, Our Question becomes :- 1/(2 - √3)

→ 1/(2 - √3)

Rationalizing The denominator we get,

→ 1/(2 - √3) * (2 + √3)/(2 + √3)

using (a + b)(a - b) = a² - b² in Denominator now,

→ (2 + √3) / [(2)² - (√3)²]

→ (2 + √3) / [ 4 - 3 ]

→ (2 + √3) / 1

→ (2 + √3) (Ans.)

Answer 2

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

$\star \: {\tt{ \frac{1}{ \sqrt{7 - 4 \sqrt{3} } } }}$

${\sf{\underline{\blue{Now}}}}$

$\implies{\tt{ \frac{1}{ \big(\sqrt{7 - 4 \sqrt{3} \big)}} }}$

$\implies{\tt{ \frac{1}{ \sqrt{7 - 2 \times 2 \times \sqrt{3} } } }}$

$\implies{\tt{ \frac{1}{ \sqrt{4 + 3 - 2 \times 2 \times \sqrt{3} } } }}$

$\implies{\tt{ \frac{1}{ \sqrt{ {(2)}^{2} + ( \sqrt{3^{2} } )- 2 \times 2 \times \sqrt{3} } } }}$

$\star \: \boxed{\tt{ \purple{ ({x} - {y})^{2} = {x}^{2} + {y}^{2} - 2xy }}}$

where x=2 and y=√3

$\implies{\tt{ \frac{1}{ \sqrt{(2 - \sqrt{3} )^{2}} } }}$

${\sf{\underline{\blue{Rationalizing}}}}$

$\implies{\tt{ \frac{1}{ (2 - \sqrt{3} )} \times \frac{(2 + \sqrt{3} )}{(2 + \sqrt{3} )} }}$

$\implies{\tt{ \frac{(2 + \sqrt{3}) }{ (2 - \sqrt{3} )(2 + \sqrt{3} )} }}$

$\star \boxed{\tt{ \purple{ {x}^{2} - {y}^{2} = (x - y)(x + y)}}}$

$\implies{\tt{ \frac{(2 + \sqrt{3}) }{ {(2)}^{2} - ( \sqrt{3} )^{2} } }}$

$\implies{\tt{ \frac{2 + \sqrt{3} }{4 - 3} }}$

$\implies \boxed{\tt{ 2 + \sqrt{3} \:Ans }}$